reserve x, y, z, s for ExtReal;
reserve i, j for Integer;
reserve n, m for Nat;
reserve x, y, v, u for ExtInt;
reserve
  D for non empty doubleLoopStr,
  A for Subset of D;
reserve K for Field-like non degenerated
  associative add-associative right_zeroed right_complementable
  distributive Abelian non empty doubleLoopStr,
  a, b, c for Element of K;
reserve v for Valuation of K;

theorem
  K is having_valuation & 0 <= v.(1.K-a) implies 0 <= v.a
  proof
    assume that
A1: K is having_valuation and
A2: 0 <= v.(1.K-a);
    1.K-a = 1.K + -a & v.-a = v.a by A1,Th20;
    hence thesis by A1,A2,Th30;
  end;
